math 233 calculus 3 - fall 2017...12.1 - three-dimensional coordinate systems question....

Math 233 Calculus 3 - Fall 2017

§12.1 - THREE-DIMENSIONAL COORDINATE SYSTEMS

§12.1 - Three-Dimensional Coordinate Systems

After completing this section, you should be able to:

• Recognize right handed and left handed coordinate systems

• Determine whether an equation describes a plane. cylinder, spheres, or otherstandard shape in 3-dimensional space

• Describe the regions of 3-dimensional space defined by inequalities

• Complete the square to rewrite equations and inequalities involving 3 variables inmore standard form

Definition. R3 means

By convention, we graph points in R3 using a right-handed coordinate system. Right-handed means:

Identify the right-handed coordinate systems.

Question. For two points P1 = (x1, y1, z1) and P2 = (x2, y2, z2) inR3, what is the distancebetween P1 and P2?

Question. What is the equation of a sphere of radius r centered at the point (h, k, l)?

Example. (# 10) Find the distance from (4,�2, 6) to each of the following:

a. The point (9,�1,�4)

b. The xy-plane (where z = 0)

c. The xz-plane

d. The x-axis

Example. Describe the region (and draw a picture).

a. |z| 2

b. x2 + z2 9

c. x2 + y2 + z2 > 2z

§12.6 QUADRIC SURFACES

§12.6 Quadric Surfaces

• Match equations of quadric surfaces with their graphs

• Describe horizontal and vertical cross-sections of surfaces and predict their shapebased on the equation of the surface.

§12.6 QUADRIC SURFACES

Match the equation to the graph using only your brain (no graphing software). Check your answer with graphing software.

A. B. C.

D. E. F.

G. H. I.

1. x2 + z2 = 4 2. 36x2 + 4y2 + 9z2 = 36 3. �x2� y2 + z = 0 4. �x2 + y2 + z = 0

5. x � y2� z2 = 0 6. �x2

� 4y2 + z2 = 0 7. x2 + y2� z2 = 1 8. �x2

� y2 + z2 = 19. �x2 + y2 + z2 = 1 10. �x2 + z = 0

§12.2 VECTORS

§12.2 Vectors

• Represent vectors with arrows, components and in terms of~i, ~j, ~k.

• Add and subtract vectors

• Find the length of a vector

• Rescale vectors to find other vectors in same direction

• Use vectors to solve problems from physics and geometry

§12.2 VECTORS

Definition. A vector is a quantity with direction and magnitude (length).

Vectors are usually drawn with arrows.

Two vectors are considered to be the same if:

Example. Which pairs of vectors are equivalent?

Definition. A scalar is another word for .A scalar does not have a direction, in contrast to a vector.

§12.2 VECTORS

Vector addition: Draw ~a +~b Vector subtraction: Draw ~a �~b

Multiplication of scalars and vectors: Draw 2~a and �3~a

§12.2 VECTORS

Vector Components

If we place the initial point of a vector ~a at the origin, then the vector can be describedby

Definition. The components of a vector are

Note. Vectors in 3-dimensional space can be described in terms of three components.

§12.2 VECTORS

Question. What are the components of the vector ~AB that starts at a point A = (3, 1)and ends at a point B = (�2, 5)?

Note. In general, the components of the vector that starts at a point A = (x1, y1) andends at a point B = (x2, y2) are:

§12.2 VECTORS

Definition. Given two vectors ~a =< a1, a2 > and ~b =< b1, b2 >, and a scalar c

~a +~b =

~a �~b =

Similarly, for three-dimensional vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >,

~a +~b =

~a �~b =

Example. < 1, 2, 3 > + < 5, 7, 12 >=

§12.2 VECTORS

Question. We have defined vector addition, subtraction, and scalar multiplicationtwice.

Why are these definitions equivalent?

§12.2 VECTORS

1. Addition is commutative

2. Addition is associative

3. Additive identity (the zero vector)

4. Additive inverses

5. Distributive property

6. Distributive property

7. Associativity of scalar multiplication

8. Multiplicative identity

§12.2 VECTORS

Definition. The length of a vector ~w =< w1,w2,w3 > is

~w|| =

Note. The length of a vector is also called:

Definition. A unit vector is a vector ...

§12.2 VECTORS

Question. How can we rescale any vector to make it a unit vector? (Rescale meansmultiply by a scalar.)

Note. Rescaling a vector to make it a unit vector is also called ...

Example. Find a unit vector that has the same direction as < 5, 1, 3 >.

§12.2 VECTORS

Definition. In 2 dimensions, the standard basis vectors are

Definition. In 3 dimensions, the standard basis vectors are

Note. Vectors can always be written in terms of

Example. Write < 3,�2, 7 > as a sum of multiples of standard basis vectors.

§12.2 VECTORS

Review. Vectors can be represented with

Example. Which arrow(s) represent(s) the vector < 2, 1 >?

Example. Write < 2, 1 > in terms of the standard basis vectors~i and ~j.

Example. Find a unit vector in the direction of < 2, 1 >.

§12.2 VECTORS

Example. Find the unit vectors that are parallel to the tangent line to y = x3 at (2, 8).

§12.2 VECTORS

Example. The 2-d unit vector ~u makes an angle of ⇡3 with the positive x-axis. Find thecomponents of ~u and write ~u in terms of~i and ~j.

Example. The 2-d vector ~v has magnitude 9 and makes an angle of 5⇡6 with the positive

x-axis. Find the components of ~v and write ~v in terms of~i and ~j.

§12.2 VECTORS

Example. Two forces act as shown. Find the magnitude of the resultant force and theangle it makes with the positive x-axis.

§12.2 VECTORS

Example. Spiderman is suspended from two strands of spider silk as shown. Find thetension in each strand of spider silk. (You will need some additional information.)

§12.3 DOT PRODUCT

§12.3 Dot Product

After completing this section, students should be able to:

• Compute the dot product of two vectors from the vector components or from themagnitudes and the angle between them

• Use the dot product to find the angle between two vectors

• Use the dot product to determine if two vectors are perpendicular

• Find the scalar and vector projections of one vector onto another

• Compute the work done by a constant force moving an object in a direction that isat an angle to the direction of the force

§12.3 DOT PRODUCT

Definition. If ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >, then the dot product of ~a and ~b isgiven by

~a ·~b =

Example. Find the dot product of < 4, 2,�1 > and < 7, 0, 5 >.

Question. Is ~a ·~b a vector or a scalar?

§12.3 DOT PRODUCT

Note. Dot product satisfies the following properties:

1. Commutative Property:

2. Distributive Property:

3. Associative Property:

4. Multiplication by ~0:

Question. What do you get when you take the dot product of a vector with itself?

§12.3 DOT PRODUCT

Definition. Dot product can also be defined in terms of magnitudes and angles:

Example. Suppose ~v and ~w meet at a 45� angle and ||~v|| = 4 and ||~w|| = 6. Find ~v � ~w.

Example. Suppose ~a � ~b = 20 and ||~a|| = 10 and ||~b|| = 4. What is the angle between ~aand ~b?

§12.3 DOT PRODUCT

Note. If ✓ is the angle between ~a and ~b, then

cos(✓) =

Example. Find the angle between the vectors < 3, 1, 2 > and < 4, 6, 1 >.

§12.3 DOT PRODUCT

Question. We have two definitions of dot product:

Are they equivalent?

§12.3 DOT PRODUCT

Review. For ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 >, what are the two ways of defining~a �~b?

Example. If ~u is a unit vector, find ~u � ~v and ~u � ~w.

§12.3 DOT PRODUCT

Question. What can be said about the angle between ~a and ~b, if

~a �~b = 0

~a �~b > 0

~a �~b < 0

Question. Is it possible to find the exact angle between two vectors if all we knowis the dot product and the magnitudes of the vectors?

§12.3 DOT PRODUCT

Example. Find the acute angle between the lines. Round your answer to the nearestdegree.

5x � y = 5, 9x + y = 6

Example. Prove that for two lines that are not horizontal or vertical, the two lines areperpendicular if and only if their slopes are ...

§12.3 DOT PRODUCT

Example. True or False:

(a) ~i · ~j = ~j ·~k = ~k ·~i = 0

(b) ~i ·~i = ~j · ~j = ~k ·~k = 1

Example. Find a unit vector that is orthogonal to both~i + ~j and~i +~k.

§12.3 DOT PRODUCT

Scalar and vector projections

Definition. The scalar projection of ~bonto ~a is given by

comp~a~b =

Definition. The vector projection of ~bonto ~a is given by

proj~a~b =

§12.3 DOT PRODUCT

Example. Find the scalar and vector projections of ~b onto ~a, where~b =< �2, 3,�6 >~a =< 5,�1, 4 >

§12.3 DOT PRODUCT

Recall: The work done by a constant force, moving an object in the direction of theforce, is:

Definition. The work done by a constant force ~F moving an object along a vector ~D(not necessarily in the direction of the force) is:

Question. How can we write work in terms of the dot product?

§12.3 DOT PRODUCT

Example. A tow truck drags a stalled car along a road. The chain makes an angle of30� with the road and the tension in the chain is 500N. How much work is done by thetruck in pulling the car 3 km?

§12.3 DOT PRODUCT

Extra Example. True or False and justify your answer.

~a �~b| ||~a|| · ||~b||

§12.3 DOT PRODUCT

Extra Example. True or False and justify your answer.

~a +~b|| ||~a|| + ||~b||

Hint: ||~a +~b||2 = (~a +~b) � (~a +~b)

§12.3 DOT PRODUCT

Extra Example. Use scalar projection to show that the distance from a point P1(x1, y1)to the line ax + by + c = 0 is

|ax1 + by1 + c|p

a2 + b2

Use this formula to find the distance from the point (�2, 3) and the line 3x� 4y+ 5 = 0.

§12.4 CROSS PRODUCTS

§12.4 Cross Products

After completing this section, students will be able to:

• Compute the cross-product of two 3-dimensional vectors from their components.

• Compute the magnitude of the cross-product from the magnitude of the vectorsand the angle between them.

• Use the right hand rule to find the direction of the cross-product.

• Use cross product to find a vector perpendicular to two other vectors.

• Use cross product to determine if two vectors are parallel.

• Use cross product to find the area of a parallelogram or triangle.

• Use properties of cross product to determine if statements about vectors are trueor false.

Definition. The cross-product of two vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 > isgiven by:

~a ⇥~b =

Example. For ~a =< 1, 2, 3 > and ~b =< 5,�1, 10 >, find ~a ⇥~b.

Properties of Cross Product:

• If ✓ is the angle between ~a and ~b, with 0 ✓ ⇡, then the length of ~a ⇥~b is givenby

~a ⇥~b|| =

• The vector ~a ⇥~b is perpendicular to ...

• The direction of ~a ⇥~b is given by ...

Example. For the two vectors shown, find ||~a ⇥ ~b|| and determine whether ~a ⇥ ~b isdirected into the page of out of the page.

Proposition. If ✓ is the angle between ~a and ~b, with 0 ✓ ⇡, then

~a ⇥~b|| = ||~a|| ||~b|| sin✓

Proposition. The vector ~a ⇥~b is perpendicular to both ~a and ~b.

Proposition. The direction of ~a ⇥~b is given by the right hand rule.

Review. The cross-product of two vectors ~a =< a1, a2, a3 > and ~b =< b1, b2, b3 > isdefined in terms of components as:

or in terms of length and direction as:

Example. Find the cross product ~u ⇥ ~w where ~u, ~v, and ~w are vectors of length 5 thatlie in the x-y plane.

True or False: If two nonzero vectors ~a and ~b are parallel, then ~a ⇥~b = ~0.

True or False: If two nonzero vectors ~a and ~b have cross-product ~a ⇥~b = ~0, then ~a and~b are parallel.

Question. How can you use cross product to:

• find a vector perpendicular to two vectors?

• determine if two vectors are parallel?

Question. Is it possible to find the exact angle between two vectors if all we know isthe cross product and the magnitudes of the vectors?

Example. Find a unit vector orthogonal to both~i + ~j and~i +~k.

Example. Use the cross product to write the area of the parallelogram in terms of ~aand ~b.

Example. Find the area of the triangle with vertices P(0, 0,�3), Q(4, 2, 0), and R(3, 3, 1).

True or False

1. ~a ⇥~b is a scalar.

2. ~a ⇥ ~a = ~0.

3. For ~a,~b , ~0, if ~a ⇥~b = ~0 then ~a = ~b.

4. ~i ⇥ ~j = ~k

5. ~a ⇥~b = ~b ⇥ ~a

6. (~a ⇥~b) ⇥ ~c = ~a ⇥ (~b ⇥ ~c)

Extra Example. Are these three vectors coplanar?

~u = 2~i + 3~j +~k

~v =~i � ~j

~w = 7~i + 3~j + 2~k

Extra Example. 1. Find all vectors ~v such that < 1, 2, 1 > ⇥ ~v =< 3, 1,�5 >.

2. Explain why there is no vector ~v such that < 1, 2, 1 > ⇥ ~v =< 3, 1, 5 >.

§12.5 LINES AND PLANES

§12.5 Lines and Planes

• Recognize equations of lines in parametric form, symmetric form, and vector form.

• Recognize equations of planes.

• Write the equation of a line given its direction and a point on the line, or given twopoints, or similar information.

• Write the equation of a plane given three points, or one point and a normal vector,or through a line and a point, or through two intersecting or parallel lines, or fromsimilar information.

• Determine if planes intersect and find lines of intersection and angles of intersec-tion.

• Determine if lines intersect, are parallel, or are skew.

• Use scalar projection to find the distance from a point to a plane, or a point toa line, or between two skew or parallel lines, or between two planes, or similarconfigurations.

Example. Is the line through (�4,�6, 1) and (�2, 0,�3) parallel to the line through(10, 18, 4) and (5, 3, 14)?

What is the equation of the line through the origin, that is parallel to the line through(�4,�6, 1) and (�2, 0,�3)?

What is the equation of the line through (�4,�6, 1) and (�2, 0,�3)?

How else could you write an equation of the line through (�4,�6, 1) and (�2, 0,�3)?

Note. The equation of a line through the point (x0, y0, z0) in the direction of the vector< a, b, c > can be described:

with the parametric equations:

or, with ”symmetric equations”:

or, with the vector equation:

Example. Find the equation of the plane through the point (�3, 2, 0) and perpendicularto the vector < 1,�2, 5 >

Note. The plane through the point (x0, y0, z0) and perpendicular to the vector < a, b, c >is given by the equation

Review. Which of these equations represents a line in 3-dimensional space?

1. 3x + 5y = 2

2. 3(x � 1) + 5(y � 3) � 4(z � 2) = 0

3. x = 7 + 4t, y = 5 � 3t, z = 7t

4. x�74 =

�y+53 = z

5. ~r(t) =< 7 + 4t, 5 � 3t, 7t >

Review. The equation of a line in 3-dimensional space can be described

with parametric equations:

with ”symmetric equations”:

with the vector equation:

where < a, b, c > represents ...

and < x0, y0, z0 > represents ...

Review. The equation of a plane in 3-dimensional space can be described by:

where < a, b, c > represents ...

and < x0, y0, z0 > represents ...

Example. Find an equation of the line though the point (3,�1, 2) and perpendicular tothe plane 4x � 6y + z = 13.

Example. Find an equation for the plane through the points (3,�1, 2), (8, 2, 4), and(�1,�2,�3).

Question. What information is enough to determine the equation for a line in the plane?

Question. What information is enough to determine the equation for a line in 3-dimensional space?

Question. What information is enough to determine the equation for a plane in 3-dimensional space?

Example. Find the line of intersection of the planes 2x � y + z = 5 and x + y + z = 1.

Example. Determine whether the lines L1 and L2 are parallel, skew, or intersecting.

L1 : x = 3 + 2t, y = 4 � t, z = 1 + 3t

L2 : x = 1 + 4s, y = 3 � 2s, z = 4 + 5s

Example. Determine whether the planes are parallel, perpendicular or neither. Ifneither, find the angle between them.

x + 2y + 2z = 1, 2x � y + 2z = 1

Extra Example. Find the distance between the point (1, 2, 3) and the plane 5x+4y�3z =10.

Extra Example. Find the distance between the skew lines

L1 : x = 3 + 2t, y = 4 � t, z = 1 + 3t

L2 : x = 1 + 4s, y = 3 � 2s, z = 4 + 5s

§13.1 VECTOR FUNCTIONS

§13.1 Vector Functions

• Define a vector valued function.

• Find the domain of a vector valued function.

• Find the limit of a vector valued function.

• Match equations of vector valued functions with their graphs by considering theprojections of the graphs onto the xy, yz, and xz planes.

• Give a vector valued equation for the intersection of two surfaces.

Definition. A vector function or vector-valued function is:

If we think of the vectors as position vectors with their initial points at the origin, thenthe endpoints of ~v(t) trace out a in R3 (or in R2).

Example. Sketch the curve defined by the vector function ~r(t) =< t, sin(5t), cos(5t) >.

Example. Consider the vector function ~r(t) =t2� t

t � 1~i +p

t + 8~j +sin(⇡t)

ln t~k

1. What is the domain of ~r(t)?

2. Find limt!1~r(t)

3. Is ~r(t) continuous on (0,1)? Why or why not?

Review. Match the vector equations with the curves.

1. ~r(t) =< t2, t4, t6 >

2. ~r(t) =< t + 2, 3 � t, 2t � 1 >

3. ~s(t) =< cos(t),� cos(t), sin(t) >

Example. At what points does the helix ~r(t) =< sin(t), cos(t), t > intersect the spherex2 + y2 + z2 = 5?

Example. Find parametric equations for the curve of intersection of the paraboliccylinder y = x2 and the top half of the ellipsoid x2 + 4y2 + 4z2 = 16.

Example. Find a vector function that represents the curve of intersection of the hyper-boloid z = x2

� y2 and the cylinder x2 + y2 = 1.

Extra Example. Consider the vector function ~r(t) = te�t~i +t3 + t

2t3� 1~j +

1. What is the domain of ~r(t)?

2. Find limt!1~r(t)

§13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS

§13.2 Derivatives and Integrals of Vector Functions

By the end of this section, students should be able to:

• Compute the derivative of a vector function.

• Compute the integral of a vector function.

• When ~r(t) represents the position of a particle at time t, explain the meaning of~r0(t), its direction, and its magnitude.

Suppose a particle is moving according to the vector equation ~r(t). How can we find atangent vector that gives the direction and speed that the particle is traveling?

Definition. The derivative of the vector function ~r(t) is the same thing as the tangentvector, defined as

d~rdt= ~r 0(t) =

If ~r(t) =< r1(t), r2(t), r3(t) >, then

~r 0(t) =

The derivative of a vector function is a (circle one) vector / scalar.

The unit tangent vector is:~T(t) =

The tangent line at t = a is:

Example. For the vector function ~r(t) =< t2, t3 >

1. Find ~r 0(1).

2. Sketch ~r(t) and ~r 0(1).

3. Find ~T(1).

4. Find the equation for the tangent line at t = 1.

Definition. If ~r(t) =< f (t), g(t), h(t) >, thenZ~r(t) dt =

a~r(t) dt =

Example. ComputeZ 2

1t~i + et~j + tet~k.

END OF VIDEO

Review. If ~r(t) = 5t2~i + sin(t)~j � 3~k, how do we compute ~r 0(t)?

Question. For a vector function ~r(t) =< r1(t), r2(t), r3(t) >, what is the limit definition of~r 0(t)?

Question. If we think of ~r(t) as a space curve, what does ~r 0(t) represent geometrically?

Question. If ~r(t) represents the position of a particle at time t,

(a) what does the direction of ~r 0(t) signify?

(b) what does the magnitude of ~r 0(t) signify?

Example. Find the tangent vector, the unit tangent vector, and the tangent line for thefollowing curves at the point given

1. ~r(t) =< t, t2, t3 > at t = 1

2. ~r(t) =< t2, t4, t6 > at t = 1

3. ~p(t) =< t + 2, 3 � t, 2t � 1 > at t = 0

Example. At what point do the curves ~r1(t) =< t, 1�t, 3+t2 > and ~r2(t) =< 3�t, t�2, t2 >intersect? Find their angle of intersection correct to the nearest degree.

Derivative rules - see textbook

• Is there a product rule for derivatives of vector functions?

• Is there a quotient rule for derivatives of vector functions?

• Is there a chain rule for derivatives of vector functions?

Example. Show that if ||~r(t)|| = c (a constant), then ~r 0(t) is orthogonal to ~r(t) for all t.

Review. If ~r(t) =< f (t), g(t), h(t) >, thenZ~r(t) dt =

a~r(t) dt =

Example. Find ~p(t) if ~p 0(t) = cos(⇡t)~i + sin(⇡t)~j + t~k and ~p(1) = 6~i + 6~j + 6~k.

Extra Example. (# 53) Show that if ~r is a vector function such that ~r 00 exists, then

[~r(t) ⇥ ~r 0(t)] = ~r(t) ⇥ ~r 00(t)

Extra Example. (#57) If ~u(t) = ~r(t) � [~r 0(t) ⇥ ~r 00(t)], show that

~u 0(t) = ~r(t) · [~r 0(t) ⇥ ~r 000(t)]

§14.1 FUNCTIONS OF TWO OR MORE VARIABLES

§14.1 Functions of Two or More Variables

• Match equations of the form z = f (x, y) to graphs of surfaces and graphs of levelcurves.

• Describe the graphs of functions of three variables w = f (x, y, z) in terms of thelevel curves f (x, y, z) = k

Example. Consider the function of two variables f (x, y) = pxy

1. What is its domain?

For the function f (x, y) = pxy ...

2. What are its level curves?

For the function f (x, y) = pxy ...

3. What does its graph look like?

END OF VIDEO

1. z = sinp

x2 + y2 2. z = 1x2+4y2 3. z = sin(x) sin(y)

4. z = x2y2e�(x2+y2) 5. z = x3� 3xy2 6. z = sin2(x) + y2

I. II. III.

IV. V. VI.

A. B. C.

D. E. F.

Functions of 3 or more variables

To visualize functions f (x, y, z) of three variables, it is handy to look at level surfaces.Example. f (x, y, z) = x2 + y2 + z2

(a) Guess what the level surfaces should look like.

(b) Graph a few level surfaces (e.g. x2 + y2 + z2 = 10, x2� y2 + z2 = 20, x2

� y2 + z2 = 30)on a 3-d plot.

Example. f (x, y, z) = x2� y2 + z2

1. Guess what the level surfaces should look like.

2. Graph a few level surfaces (e.g. x2� y2 + z2 = 0, x2

� y2 + z2 = 10, x2� y2 + z2 = 20)

on a 3-d plot.

§14.2 LIMITS AND CONTINUITY

§14.2 Limits and Continuity

1. Use a graph to build intuition for whether or not a limit of a function of twovariables exists.

2. Prove that a limit of a function of two variables does not exist by finding two pathsalong which the function approaches di↵erent values.

3. Prove that a limit of a function of two variables does exist by converting to polarcoordinates and using the squeeze theorem.

4. State the definition of continuity for functions of two variables in terms of limits.

5. Determine where a function is continuous.

Recall LIMITS from Calculus 1:

Informally, limx!a

f (x) = L if the y-values f (x) get closer and closer to the same number Lwhen x approaches a from either the left of the right.

Does limx!0

f (x) exist for these functions?

LIMITS for functions of two variables:

Informally, lim(x,y)!(a,b)

f (x, y) = L if the z-values f (x, y) ...

For each function, decide if lim(x,y)!(0,0)

f (x, y) exists. The color is based on height: low is

blue and high is red.

A) lim(x,y)!(0,0)

x2� y2

x2 + y2 B) lim(x,y)!(0,0)

1 � x2

x2 + y2 + 1

C) lim(x,y)!(0,0)

xx2 + y2 D) lim

(x,y)!(0,0)

x2 + y2

Example. lim(x,y)!(0,0)

x2� y2

x2 + y2

1 � x2

x2 + y2 + 1

xx2 + y2

x2 + y2

END OF VIDEO

Review. Informally, lim(x,y)!(a,b)

f (x, y) = L if the z-values f (x, y) ...

lim(x,y)!(0,0)

x2� y2

x2 + y2

lim(x,y)!(0,0)

x2 + y2

Does lim(x,y)!(0,0)

f (x, y) exist? The color is based on height: low is blue and high is red.

A) lim(x,y)!(0,0)

2xyx2 + y2 B) lim

(x,y)!(0,0)

x2 + y4

C) lim(x,y)!(0,0)

10 sin(xy)x + y + 20

D) lim(x,y)!(0,0)

3x2y � 3x2� 3y2

x2 + y2

Example. Show that lim(x,y)!(0,0)

2xyx2 + y2 does not exist.

x2 + y4 does not exist.

f (x, y) =10 sin(xy)x + y + 20

does exist.

3x2y � 3x2� 3y2

x2 + y2 does exist.

Summary: for practical purposes, the best way to show that a limit does not exist is to:

For practical purposes, the best ways to show that a limit exists is to:

Recall CONTINUITY from Calculus 1: A function f is continuous at the point x = a if:

Recall from Calculus 1: common functions that are continuous include:

CONTINUITY for functions of of two (or more) variables:

A function f (x, y) is continuous at the point (x, y) = (a, b) if:

Common functions that are continuous include:

Example. Where is f (x, y) =

4 � x2 +p

4 � y2

1 � x2� y2 continuous?

§14.3 PARTIAL DERIVATIVES

§14.3 Partial Derivatives

• Compute partial derivatives.

• Use average rates of change to approximate partial derivatives.

• For functions of two variables, explain the geometric meaning of a partial deriva-tive as the slope of a tangent line to a curve in the intersection of a surface and aplane.

Example. (from book) The wave heights h in the open sea depend on the speed ⌫ ofthe wind and the length of time t that the wind has been blowing at that speed. So wewrite h = f (⌫, t).

1. What is f (40, 20)?

2. If we fix duration at t = 20 hours andthink of g(⌫) = f (⌫, 20) as a functionof ⌫, what is the approximate valueof the derivative dg

��⌫=40

3. If we fix wind speed at 40 knots, andthink of k(t) = f (40, t) as a function ofduration t, what is the approximatevalue of the derivative dk

��t=20?

Definition. For a function f (x, y) defined near (a, b), the partial derivatives of f at (a, b)are:

fx(a, b) = the derivative of f (x, b) with respect to x when x = a, and

fy(a, b) = the derivative of f (a, y) with respect to y when y = b.

In terms of the limit definition of derivatives, we have:

fx(a, b) = limh!0

fy(a, b) = limh!0

Geometrically, f (x, b) can be thought of as

So fx(a, b) = ddx f (x, b)|x=a can be thought of as

Note. To compute fx, we just take the derivative with x as our variable, holding allother variables constant. Similarly for the partial derivative with respect to any othervariable.

Example. f (x, y) = xy. Find fx(1, 2) and fy(1, 2).

Notation. There are many notations for partial derivatives, including the following:

fx@ f@x

@z@x f1 D1 f Dx f

Note. Partial derivatives can also be taken for functions of three or more variables. Forexample, if f (x, y, z,w) is a function of 4 variables, then fz(3, 4, 2, 7) means:

END OF VIDEO

Review. Which of the following is the limit definition of@ f@x

(1, 2)?

A. limh!0

f (1 + h, 2 + h) � f (1, 2)h

B. limh!0

f (1 + h, 2) � f (1, 2 + h)h

C. limh!0

f (1 + h, 2) � f (1, 2)h

D. limh!0

f (1, 2 + h) � f (1, 2)h

Review. Based on the graph of z = f (x, y) shown

• is@ f@x

(1, 2) positive, negative, or 0?

• is@ f@y

(1, 2) positive, negative, or 0?

Question. For f (x, y) = 3x2y + 5y2 + cos(xy), find@ f@x

Example. For f (x, y, z) = ex sin(y�z), find Dx f and Dz f .

Example. For z = f (x)g(y), find an expression for@z@x

Example. For x2� y2 + z2

� 2z = 12, find@z@y

Notation. Second derivatives are written using any of the following notations:

Example. For f (x, y) = x2 + x2y2� 2y2, calculate fxx, fxy, fyx, and fyy.

Theorem. (Clairout’s Thm) Let f (x, y) be defined on a disk D containing (a, b). If fxy

and fyx are both defined and on D, then fxy(a, b) = fyx(a, b).

Example. Consider the function

f (x, y) =

8>><>>:xyx2�y2

x2+y2 if (x, y) , (0, 0)

0 if (x, y) = (0, 0)

It turns out that

• for all x D2 f (x, 0) =

• for all y D1 f (0, y) =

Therefore,

• D2,1 f (0, 0) =

• D1,2 f (0, 0) =

Question. True or False: There is a function f with continuous second partial deriva-tives such that fx = x and fy = x.

Extra Example. The wave equation is given by

@2u@t2 = a2@

where u(x, t) represents displacement, t represents time, x represents the distance fromone end of the wave, and a is a constant.

Verify that the function u(x, t) = sin(x � at) satisfies the wave equation.

§14.4 TANGENT PLANES AND LINEAR APPROXIMATIONS

§14.4 Tangent Planes and Linear Approximations

• Find the equation for the tangent plane of a surface z = f (x, y) at a point.

• Use the tangent plane to approximate a function.

• Compute the di↵erential and use it to estimate errors.

• Explain the relationship between the tangent plane, the linearization equation, andthe di↵erential.

• Compute the tangent plane from information about the function’s value and partialderivative values.

• Use linearization to interpolate between numerical values in a 2 x 2 talbe.

Tangent line:

Tangent plane:

Example. Find the tangent plane to the surface z = y2� x2 at the point (1, 2, 3).

Find a general formula for the tangent plane of z = f (x, y) at the point (x0, y0, z0).

Notice that this formula is analogous to the formula for a tangent line for a function ofone variable.

Usually, the tangent plane at a point (x0, y0, z0) is a good approximation of the surfacenear that point.

Example. Find the tangent plane for the surface z = f (x, y) when (x, y) = (1, 1), wheref (x, y) = 1 � xy cos(⇡y). Use it to approximate f (1.02, 0.97).

END OF VIDEO

Example. Suppose we have a surface z = f (x, y) and we know that f (1, 2) = 6, fx(1, 2) =�7 and fy(1, 2) = 4. What is the equation for the tangent plane to the surface at thepoint (1, 2, 6)?

Example. Find the tangent plane for the surface z = sin(x)+ y2 at (x, y) = (0, 2) and useit to approximate sin(0.2) + (1.9)2

This method of approximating a function’s value with the height of the tangent planecan be written in terms of a linearization equation or in the language of di↵erentials.

Definition. The linearization of f (x, y) is written as:

Recall: In Calc 1, for a function y = f (x), the di↵erential was defined as d f = f 0(x)dx.

Definition. The di↵erential for the function y = f (x, y) is:

� f represents the actual change in a function f (x, y) � f (a, b). d f represents the corre-sponding change in the tangent plane between (a, b) and (x, y).

Di↵erentials are useful for estimating errors and making approximations, because thedi↵erential is linear in all its variables, and linear functions are easier to calculate.

Example. Use di↵erentials to estimate the amount of metal in a closed cylindrical canthat is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cmthick and the metal in the sides is 0.05 cm thick.

Example. Four positive numbers, each less than 50, are rounded to the first decimalplace and multiplied together. Use di↵erentials to estimate the maximum possibleerror in the computed product that might result from the rounding.

When the tangent plane approximation works and when it doesn’t.

Note. Anytime fx and fy exist at a point (a, b), it is possible to write down an equationfor a tangent plane at the point where (x, y) = (a, b).

• Usually ...

• Sometimes ...

Definition. (Informal definition) A function is f is called di↵erentiable at (a, b) if ...

Theorem. If fx and fy exist in near (a, b) and are at (a, b), then f isdi↵erentiable at (a, b).

Example. Here is an example of a function that is not di↵erentiable at (x, y) = (0, 0).

f (x, y) =

8>><>>:xy

x2+y2 if (x, y) , (0, 0)

0 if (x, y) = (0, 0)

Question. In what sense is the tangent plane not a good approximation to the function?

Question. Is this function continuous at (0, 0)?

Question. What was the relationship between di↵erentiability and continuity in Cal-culus 1?

Question. What is the relationship between di↵erentiability and continuity for func-tions of several variables?

Recap: The tangent plane, the linearization, and the di↵erential are related as follows:

§14.5 THE CHAIN RULE

§14.5 The Chain Rule

After completing the section, students should be able to:

• Compute partial derivatives using the Chain Rule, using formulas for functions,or using numerical information such as tables of values or contour graphs.

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